(b43i 


IC  28  19U 


GEOMETRIC  PROPERTIES  COMPLETELY 
CHARACTERIZING  ALL  THE  CURVES 
IN  A  PLANE  ALONG  WHICH  THE  CON- 
STRAINED MOTION  IS  SUCH  THAT  THE 
PRESSURE  IS  PROPORTIONAL  TO  THE 
NORMAL  COMPONENT  OF  THE  ACTING 
FORCE. 


BY 


SARAH  ELIZABETH   CRONIN 


f       o-ftu-b; 

'    DNIVSRSXXy 


Submitted  in  Partial  Fulfilment  of  the  Requirements  for 

THE  Degree  of  Doctor  of  Philosophy,  in  the  Faculty 

OF  Pure  Science,  Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1917 


GEOMETRIC  PROPERTIES  COMPLETELY 
CHARACTERIZING  ALL  THE  CURVES 
IN  A  PLANE  ALONG  WHICH  THE  CON- 
STRAINED MOTION  IS  SUCH  THAT  THE 
PRESSURE  IS  PROPORTIONAL  TO  THE 
NORMAL  COMPONENT  OF  THE  ACTING 
FORCE. 


BY 

SARAH  ELIZABETH   CRONIN 


Submitted  in  Partial  Fulfilment  of  the  Requirements  for 

THE  Degree  of  Doctor  of  Philosophy,  in  the  Faculty 

OF  Pure  Science,  Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1917 


CONTENTS. 

Introduction. 

Chapter  I.    The  Differential  Equation  of  the  System 1 

Chapter  II.     Geometric  Properties  of  the  System ,  4 

Section  1.  Osculating  Conies 4 

2.  Curvature  of  the  Conic  determined  by  Theorem 

1 5 

3.  Second  Center  of  Curvature  of  the  Conic  deter- 
mined by  Theorem  1 5 

4.  Tangents  to  the  Parabola  determined  by  The- 
orem III 6 

5.  Center  of  the  Conic  determined  by  Theorem  I . .  7 

6.  Hyperosculating  Parabolas 8 

7.  Inverse  Curve 10 

8.  Circles  of  Curvature  Hyperosculating  the  Conies 
determined  by  Theorem  1 10 

9.  Mid-point  of  the  Chord  of  the  Inverse  Curve 
cut  from  the  Normal  to  the  Lineal  Element 11 

10.  Osculating  Circles  having  Five-point  Contact . .  12 

Chapter  III.    Conversion  of  the  Properties  stated  in  The- 
orems I-IX 14 

Section  1.  Conversion  of  Property  1 14 

II 14 

III 15 

IV 16 

V 17 

VI 18 

VII 20 

VIII 21 

IX 23 

Chapter  IV.    Complete  Characterization 25 

Section  1.  Relations  existing  among  the  Arbitrary  Func- 

iii 


*\  i'^  y^  i\  (-x  ■  - 


2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

IV  CONTENTS. 

tions  entering  into  the  Differential  Equation  of 
the  System  of  Curves  having  Properties  I-IX 
if  it  is  to  be  of  the  Special  Form  given  in  Chapter 

1 25 

2.  Geometric  Interpretations  of  the  Relations  of 
Section  1 27 


INTRODUCTION. 

In  his  "  Differential-Geometric  Aspects  of  Dynamics,"* 
Kasner  gives  a  set  of  five  geometric  properties  which  completely 
characterize  a  system  of  curves  in  a  plane  field  of  force  {(p,  yp), 
along  which  the  constrained  motion  of  a  particle  is  such  that 
the  pressure  exerted  by  the  particle  on  the  curve  is  a  given 
multiple  of  the  normal  component  of  the  acting  force.  The 
defining  equation  of  the  system  is  P  =  kN,  when  P  is  the  pres- 
sure, N  the  normal  component  of  the  force,  and  k  a  given  con- 
stant. 

Denoting  the  tangential  component  of  the  force  by  T,  the 
equations  of  motion  are 


where 


-=T     P^-'-N^kN 

^IT+Y"  VrrP'  y"      ' 


If  at  the  initial  time  to  the  particle  is  at  the  point  (ar©,  yo) 
moving  in  the  direction  yo'  with  the  speed  Vq,  the  path  of  the 
particle  is  uniquely  determined.  By  varying  the  initial  direc- 
tion 2/0'  and  the  initial  speed  Vq,  we  obtain  00  2  curves  of  the 
system  through  the  given  point  (xo,  yo).  Since  the  path  of  the 
particle  is  determined  by  taking  any  one  of  its  00  ^  points  as  the 
initial  point,  the  system  contains  in  all  00  ^  curves. 

The  differential  equation  of  the  system  in  intrinsic  form, 
obtained  by  eliminating  v  between  dv/dt  =  T  and  v^jr  =  (A  +  l)N, 
is  _ 

Nr,=  {n+l)T-rN,  (1) 

where 


The  equivalent  equation  in  Cartesian  coordinates  is  of  the 
third  order  and  represents,  for  every  value  of  k,  a  triply  infinite 
*  Princeton  Colloquium  Lectures,  page  91. 


VI  INTRODUCTION. 

system  denoted  by  Sk-  The  triply  infinite  systems  of  trajectories, 
brachistochrones,  catenaries  and  velocity  curves  are  but  special 
cases  of  this  more  general  system,  being  obtained  from  it  for 
particular  values  of  k  as  follows: 

Sq,  the  system  of  trajectories. 
Si,  the  system  of  brachistochrones, 
Si,  the  system  of  catenaries, 
(Soo,  the  system  of  velocity  curves. 

In  a  given  field  of  force,  there  are  oo^  of  these  triply  infinite 
systems  corresponding  to  the  oo^  values  of  k.  The  elimination 
of  k  from  the  differential  equation  (1)  gives  the  differential 
equation  of  the  quadruply  infinite  system  obtained  by  com- 
bining all  the  systems  Sk  in  the  given  field.  The  intrinsic  form 
of  this  differential  equation  is 


where 


Nts  +  rN  T 


0, 


It  is  the  purpose  of  the  present  paper  to  obtain  a  complete 
characterization  of  this  oo^  system  of  curves.  The  differential 
equation  of  the  system  in  Cartesian  form  is  obtained  in  Chapter  I. 
Geometric  properties  of  the  system  are  derived  in  Chapter  II. 
At  any  point  (x,  y)  of  the  plane,  there  are  oo  ^  curves  of  the  family 
having  a  given  direction  y'  and  a  given  radius  of  curvature  r. 
If  the  osculating  conic  of  each  of  these  curves  is  constructed,  the 
locus  of  the  centers  of  these  conies  is  a  conic  passing  through 
the  given  point  in  the  given  direction.  Applying  this  property 
to  the  "  four  special  cases  of  physical  interest,"  Kasner  restates 
it  as  follows:  "  In  any  plane  field  of  force  select  any  fixed  element 
of  curvature,  corresponding  to  the  initial  values  x,  y,  y',  r  so 
given,  construct  the  unique  trajectory,  unique  brachistochrone, 
unique  catenary,  the  unique  velocity  curve,  and  the  respective 
centers  of  the  osculating  conies;  the  four  centers  so  formed  and 
the  given  point  {x,  y)  will  lie  on  a  conic  passing  through  the 


INTRODUCTION.  Vll 

latter  point  in  the  given  direction  y' "    The  radius  of  curvature 
of  this  conic  is  r/2. 

The  00  2  curves  of  the  family  at  the  given  point  {x,  y)  in  the 
given  direction  y'  have  associated  with  them  oo  i  of  these  conies. 
The  centers  of  these  conies  lie  on  a  conic  passing  through  the 
given  point,  and  their  second  centers  of  curvature  lie  on  a  parab- 
ola passing  through  the  given  point  and  having  its  axis  parallel 
to  the  given  direction.  The  angles  (p,  <pi  and  6  which  the  conic, 
the  parabola  and  the  direction  of  the  acting  force,  respectively, 
make  with  the  given  direction  are  related  as  follows: 

tan  <p  =  sin  20, 

3  tan  <pi  =  —  sin  26. 

One  of  the  oo^  conies  associated  with  the  <»2  curves  of  the 
family  at  the  point  {x,  y)  in  the  direction  y'  has  four  point  contact 
with  its  osculating  circle  at  {x,  y).  The  centers  of  the  « ^  hyper- 
osculating  circles  thus  associated  with  all  the  curves  of  the  family 
through  {x,  y)  lie  on  a  cubic  which  has  as  tangents  the  minimal 
lines  at  the  given  point. 

Of  the  00 1  curves  of  the  family  corresponding  to  a  given  curva- 
ture element  {x,  y,  y',  r)  there  are  two  which  are  hyperosculated 
by  their  osculating  parabolas.  The  locus  of  the  foci  of  these 
hyperosculating  parabolas  as  r  varies  in  a  bicircular  quartic 
with  a  node  at  the  given  point.  One  of  the  nodal  tangents  is 
perpendicular  to  the  direction  of  the  acting  force  and  the  other 
makes  with  this  direction  an  angle  which  is  bisected  by  the  given 
direction. 

The  inverse  of  this  quartic  is  an  hyperbola  with  asymptotes 
parallel  to  the  nodal  tangents  of  the  quartic.  The  locus  of  the 
mid-point  of  the  chord  of  this  hyperbola  cut  from  the  normal  to 
the  given  direction  at  the  given  point  is  a  circular  cubic  which 
has  a  double  point  at  the  given  point  and  its  real  asymptote 
parallel  to  the  direction  of  the  acting  force. 

One  of  the  oo^  curves  of  the  family  corresponding  to  a  given 
lineal  element  {x,  y,  y')  has  five-point  contact  with  its  osculating 
fircle.  The  locus  of  the  center  of  this  five-point  circle  as  y' 
varies  is  a  conic  passing  through  the  given  point.  Its  tangent 
at  this  point  is  perpendicular  to  the  direction  of  the  acting  force. 


Vm  INTRODUCTION. 

If  a  quadruply  infinite  system  of  curves  in  a  plane  is  given  at 
random,  there  exists  in  general  no  field  of  force  in  which  the 
pressure  along  the  curves  of  this  system  is  proportional  to  the 
normal  component  of  the  force.  The  question  of  the  suflficiency 
of  these  properties  to  insure  the  existence  of  such  a  field  of  force 
is  next  investigated,  in  Chapter  III. 

It  is  found  that  these  properties  belong  to  other  quadruply 
infinite  systems  of  curves  besides  the  system  of  curves  along 
which  P  =  kN,  and  hence  are  not  suJ0Bcient  for  a  complete 
characterization. 

Additional  properties  are  obtained  in  Chapter  IV  which 
complete  the  characterization. 

The  entire  set  of  sixteen  properties  obtained  is  shown  to  be 
both  necessary  and  sufficient  for  a  complete  characterization. 

The  writer  is  indebted  to  Professor  Kasner  for  helpful  sug- 
gestions and  criticisms. 


CHAPTER  I. 

DiFPEKENTIAL  EQUATION   OF  THE  SySTEM. 

We  consider  the  constrained  motion  of  a  particle  in  a  plane 
moving  along  a  curve  under  the  action  of  a  positional  force;  that 
is,  a  force  whose  rectangular  components  are  functions  only  of 
the  coordinates  of  the  point  at  which  it  acts,  namely  ip{x,  y) 
and  xpixy  y).  Denoting  by  Sk  the  system  of  curves  defined  by 
the  equation  P  =  kN  for  a  given  value  of  k,  P  being  the  pressure 
and  iV  the  normal  component  of  the  acting  force,  the  problem  is 
to  obtain  geometric  properties  completely  characterizing  the 
curves  of  all  the  systems  Sk^ 

The  differential  equation  of  the  system  Sk  is 

Nr,=  (n-\-l)T-rN,  (1) 

where 

^~  k+1'    ^^  ~  l  +  y"  ^"^^ 

and  T  is  the  tangential  component  of  the  acting  force.  Differ- 
entiating equation  (1)  with  respect  to  S,  and  eliminating  n,  we 
obtain  the  differential  equation,  in  an  intrinsic  form,  of  all  the 
systems  Sk  to  be 


where 


Nts  +  rN  T 


^  ~  l  +  y" 


(3) 
(4) 


JTo  obtain  the  equivalent  of  (3)  in  Cartesian  coordinates,  we 
substitute  for  N  and  T  the  values  given  in  (2)  and  (4),  and  for 
N,  T,  r,  Tg,  Tas  the  values  given  by 

1 


CURVES  IN  A  PLANE. 


^  ,     (1  +  y")y"' 


2/' 


_L,,      ..  [(1  +  y")y'^ + 2y'y"y"'\  -  2(i  +  y")y""  \ 
"r  "^  y'"  i 


(5) 


1 

X 


Making  these  substitutions  in  (3),  we  obtain 

,  r_3 3/(^^+^)        1  3 

■*"  L 1  +  2/"      (^  -  2/V)(^  +  y'^)(l  +  2/'')  J  ^ 

L  (^  +  2/V)(iA-2/V)  J^  • 

Introducing 

Pi  =  ^x  +  (^»  -  <p^y'  -  <pvy''^y 
P2=  <Px+  {<Pv  +  ^r)y'  +  M\ 
^+^ 2y' 

^2-^  +  y'^      ^-2/V'  ^^ 

„ 3 3y^(^^+^) 

^'  ~  1  +  2/''     (^  -  y'<p){v>  +  2/'^)(l  +  2/'') ' 

QP^y' ^P^y'  +  Pi 

^'  "  (^  -  2/V)(l  +  y")      (^  +  2/'^)(l  +  2/") ' 


{<P  +  y'rP){rl^-y'<p) 


THE  DIFFERENTIAL  EQUATION  OP  THE  SYSTEM.  3 

(6)  may  be  written 

y^  =  |-/  y""  +  iQ^y"  +  Q-^)y"'  +  Q^y'"  +  Q^y"''  +  Q^y"-   («) 

This  is  a  differential  equation  of  the  fourth  order  and  rep- 
resents a  quadruply  infinite  system  of  curves. 


CHAPTER  II. 

Geometeic  Properties  of  the  System. 

Section  1.    Osculating  Conies. 

Consider  the  curves  of  this  system  through  a  given  point 
(x,  y)  of  the  plane,  in  a  given  direction  y',  with  a  given  curvature. 
These  form  a  system  of  oo^  curves,  all  of  which  have  the  same 
first  center  of  curvature,  but  each  of  which  corresponds  to  a 
different  second  center  of  curvature.  Each  of  these  curves  has  a 
definite  osculating  conic  at  the  given  point;  that  is,  a  conic 
having  fourth  order  contact  with  the  curve.  The  coordinates 
of  the  center  of  the  osculating  conic,  if  the  given  point  is  taken 
as  the  origin  of  coordinates,  are 

^y"y"'  ,  ^  ^y'y"y"'  -  ^y'" 

5y"'^-dy"y^'  5y'"^  -  Sy'Y 


The  locus  of  the  centers  of  all  the  osculating  conies  correspond- 
ing to  the  00^  curves  of  the  system  is  obtained  by  eliminating 
y^^  and  y'"  between  equations  (8)  and  (9). 

From  (9)  we  have 

y     "y'h-k*     y     ~  y'h-k\_y'h-k      h]' 
Substituting  these  values  in  (8)  and  reducing,  we  obtain 

[(3  +  3Qi  y'  +  qzy")y"'  +  m^v'  +  Q^y")y"'  +  Q^y'YW 

-  [(3Qi  +  2Q^y')y'"  +  (3^2  +  2Q,y')y"'  +  2Q,yY]hk    (10) 

+  [Qzy'"  +  Q^y'"  +  Q^y"W  +  ^yY'h  -  Sy'"k  =  o. 

This  is  the  equation  of  a  conic  passing  through  the  given 
point;  the  equation  of  its  tangent  at  the  origin  is  k  —  y'h  =  0. 

Theorem  I.  7f,  for  each  of  the  oo  ^  curves  of  the  system  corre- 
sponding to  a  given  curvature  element,  we  construct  the  osculating 
conic,  the  locus  of  the  centers  of  these  conies  is  a  conic  passing  through 
the  given  point  in  the  given  direction. 

4 


GEOMETEIC  PEOPERTIES  OP  THE  SYSTEM. 

Section  2.    Curvature  of  the  Con,ic  of  Theorem  L 

Applying  the  formula 

^kfdh^ 
K  — 


[1  +  {dkldhff^ 

to  obtain  the  curvature  at  the  given  point  of  the  conic  (10),  we 
find 

Theorem  II.  The  curvature  at  the  given  point  of  the  conic 
corresponding,  by  theorem  I,  to  a  given  curvature  element  is  twice 
the  curvature  of  the  element. 

Section  3.    Second  Center  of  Curvature  of  the  Conic  Determined  by 

Theorem  I, 
For  the  conic  (10),  we  find 

The  coordinates  of  the  second  center  of  curvature  are  ob- 
tained by  the  formulas 


-  4(1  +  y")y'  ,  {i  +  y'*)\.„ 

Applying  these  to  the  conic  (10),  we  get 
2(1  +  y")y'     3(1+Y!)! 

(1  +  y'^){\  -  Sy")     3  (1  +  y")y'{q,y"  +  Q^) 
2/2  -  y„  4  y//a 


(11) 


(11') 


We  now  consider  all  curves  of  the  system  corresponding  to  a 
given  lineal  element  {x,  y,  y').  These  form  a  system  of  oo^ 
cm-ves.  In  accordance  with  theorem  I,  there  are  ooi  conies 
associated  with  this  system  of  oo^  curves,  each  conic  being  as- 
sociated, for  different  values  of  y",  with  the  oo  ^  curves  corre- 
sponding to  a  curvature  element  {x,  y,  y',  y"). 


6  CURVES  IN  A  PLANE. 

The  locus  of  the  second  centers  of  curvature  of  these  conies, 
with  the  given  point  taken  as  origin,  is  obtained  by  eliminating 
y"  between  equations  (11). 

Multiplying  the  first  of  these  equations  by  y'  and  then  sub- 
tracting from  the  second,  we  have 

J_^  2(y2  —  y'xj) 
y"        (1  +  2/")'   * 
Substituting  this  in  the  first  of  the  equations  (11),  we  have, 
after  reduction, 

6^2(2/2  -  y'x^y  +  (1  +  2/")[3(l  +  y'^Qx  +  %y']yi 

+  (1  +  y")[2  -  Qy''  -  3Qi(l  +  y")y']x,  =  0.     ^^^^ 

This  is  the  equation  of  a  parabola  passing  through  the  given 
point;  the  axis  of  the  parabola  is  parallel  to  the  line  2/2  =  y'x^. 

Theorem  III.  The  locus  of  the  second  centers  of  curvature  of 
the  00 1  conies  associated,  by  theorem,  I,  with  a  given  lineal  element 
(x,  y,  y')  is  a  parabola  passing  through  the  given  point;  the  axis  of 
the  parabola  is  parallel  to  the  given  direction. 

Section  4'     The  Equation  of  the  Tangent  to  the  Parabola. 
The  equation  of  the  tangent  to  the  parabola  at  the  given  point  is 
_  2-62/-'-3Qi(l  +  y-V^  .^„ 

If  we  substitute  for  Qi  its  value  given  in  (7)  and  take  the  x 
axis  in  the  direction  y',  we  obtain 

± 

slope  of  tangent  =  ~  o p  •  (14) 

1  +  72 

Calling  the  angle  this  tangent  makes  with  the  given  direction 
6\,  and  the  angle  the  acting  force  makes  with  the  given  direction 
di,  (14)  may  be  written 

tan  ^1  =  —  ^  sin  202-  (15) 

Theorem  IV.  At  the  given  point,  the  tangent  to  the  parabola 
makes  with  the  given  direction  an  angle  whose  tangent  is  to  the  sine 
of  twice  the  angle  the  given  direction  makes  with  the  acting  force  as 
1  is  to  3. 


GEOMETRIC  PROPERTIES  OF  THE  SYSTEM. 

Section  6.    Center  of  the  Conic  of  Theorem  I. 
The  coordinates  of  the  center  of  the  conic 

ax^  +  bxy  -\-  cy^  -{-  dx -{-  ey  =  0 
are  given  by  the  formulas 

2cd  —be       „      2ae  —  bd 


a 


/3  =  iJ—JZT'  (16) 


6^  —  4ac '  b^  —  4ac  * 

Applying  these  formulas  to  conic  (10), 

3(to"  +  Q2) ,.-. 

"  -      (3Qi2  -  4Qz)y'"+  {QQ1Q2  -  4.Q,)y"+(dQ2'  -  4.Q,) '    ^''^ 

P  ~      (3Qi2  -  4^3)2/"'  +  {QQ1Q2  -  ^Q4)y"  +  (3^2^  -  4Q6)  *    ^  ""^ 

The  locus  of  the  centers  of  the  <»  ^  conies  associated,  by  the- 
orem I,  with  the  00^  curves  corresponding  to  a  given  lineal 
element  is  obtained  by  eliminating  y"  between  equations  (17)  and 
(18).    To  carry  out  this  elimination,  we  take 

0  _i2-^Qiy')y"+Q2y' 

a  Qiy"-\-Q2 

from  which 

„  _     Q2W  -  y'ot) 

y        2a-  Q,W  -  y'cc) ' 
Substituting  this  value  of  y"  in  (17),  we  obtain 
2[(Q2^Q3  -  Q1Q2Q4  +  Qm,)y"  -  2(^2^4  -  2qmy' 

-  (3Q22  -  4Q6)]a2  _  4[(Q2^Q3  -  Q1Q2Q4  +  Qi^q,)y' 

-  (Q2Q4  -  2QiQ5)]a^  +  2[Q2=^Q3  -  Q1Q2Q4  +  q^Q,,W 

-  3[2Q2  +  QiQ22/']a  +  3Q1Q2/3  =  0. 

This  is  a  conic  passing  through  the  given  point.  The  equa- 
tion of  the  tangent  at  the  given  point  is 

,      2  +  Qiy' 

If  we  substitute  for  Qi  its  value  from  (7),  and  take  the  x  axis 
in  the  direction  y' , 


O  CUKVES  IN  A  PLANE. 

the  slope  of  the  tangent  = jp .  (20) 

Callmg  the  angle  this  tangent  makes  with  the  given  direction 
Oi,  and  the  angle  the  acting  force  makes  with  the  given  direction 
62,  (20)  may  be  written, 

tan  di  =  sin  2^2-  (21) 

Theokem  V.  The  locus  of  the  centers  of  the  <x>i  conies  asso- 
ciated, by  theorem  I,  with  a  given  lineal  element  is  a  conic  passing 
through  the  given  point.  The  tangent  to  this  conic  at  the  given  point 
makes  with  the  given  direction  an  angle  whose  tangent  is  equal 
to  the  sine  of  twice  the  angle  the  acting  force  makes  with  the  given 
direction. 

Section  6.    Hyperosculating  Parabolas. 

We  now  determine  all  the  curves  of  the  system  through  the 
point  {x,  y)  which  are  hyperosculated  by  their  osculating  parab- 
olas.   The  differential  equation  of  all  parabolas  in  the  plane  is 

The  hyperosculating  parabola  has  contact  of  the  fourth  order 
with  its  curve,  hence  its  derivatives  up  to  the  fourth  order  at 
the  given  point  are  the  same  as  those  of  its  curve.  Substituting 
the  value  of  y^  from  (22)  in  equation  (8),  we  find 

y""+  Sy"(Qiy"  +  ^2)2/'"+  Sy"iQ,y"'+  Q^'"+  Q,y")  =  0.    (23) 

This  is  an  equation  of  the  second  degree  in  y'".  Hence  two 
curves  of  the  system  of  00  ^  curves  corresponding  to  a  given  curv- 
ature element  are  hyperosculated  by  their  osculating  parabolas. 

The  coordinates  of  the  focus  of  the  osculating  parabola, 
referred  to  the  given  point  {x,  y)  as  origin,  are  given  by  the 
formulas 


a  = 


^=- 


dyV  W"(y"  -  1)  +  2y'{Sy"  -  y'y'")] 
2  y""-\-(^y"'-y'y"r        ' 

^y"  [(3y  "  -  y'y"')iy"-  D  -  2yy^1 

///2  _i_  (o,,n2 „,/„"'\2 


y""  +  W  -  y'y'") 


GEOMETRIC  PROPERTIES  OP  THE  SYSTEM. 


(26) 


From  these  we  have 

(ay'  -  )8)(1  +  y'') 

(25) 

,„  ^  _  3(a2/^  -  ^)(1  +  y")[2^y'  +  a(l  -  y'^\ 
y  4(a2  4-  (82)2 

To  obtain  the  locus  of  the  focus  of  the  hyperosculating  parabola 
as  the  curvature  of  the  element  changes,  its  direction  remaining 
unchanged,  we  eliminate  y"  and  y'"  between  (23)  and  (25). 
The  result  of  this  elimination,  after  substituting  the  values  of 
Qu  Qi,  Qs,  Qi,  Qb,  given  in  (7),  and  reducing,  is 

4PiP2(a2+/3T+{[12Pi(^+2/V)-2(3P2+Piy')(^-2/V)]« 

-  [(6P22/'- 2Pi)  (,A- 2/V)  -  12Pi2/'(^+2/V)]i8 }  («'+^') 

+m+y")[(<pW+2v>Y-  <P^W-  (^+tA')(l-2/'')«/3 

-^(<p4^y''-2rPY-cpxpm  =  0. 

This  is  the  equation  of  a  bicircular  quartic  having  a  node  at 
the  given  point.    The  equations  of  the  tangents  at  the  origin  are 

/3[(1 -/')?'+ 2y VI  +  «[(1  -  !/'V  -  2!/Vl  =  0. 

Each  of  these  tangents  has  second  order  contact  with  the 
quartic  at  the  given  point,  cutting  it  in  three  coincident  points, 
two  on  one  branch  and  one  on  the  other. 

If  we  take  the  x  axis  in  the  direction  y',  the  equations  of  the 
tangents  reduce  to 

^xl^-\-a<p=  0, 

0(p  -\-  a\l/  =  0. 

Hence  the  first  tangent  is  perpendicular  to  the  direction  of 
acting  force,  and  the  second  tangent  makes  with  this  direction 
an  angle  which  is  bisected  by  the  given  direction. 

Theorem  VI.  Corresponding  to  each  curvature  element,  there 
are  two  curves  of  the  system  that  are  hyperosculated  by  their  osculating 
parabolas.  The  hcus  of  the  foci  of  these  hyperosculating  parabolas 
as  the  curvature  of  the  element  varies,  the  direction  of  the  element 
remaining  unchanged,  is  a  bicircular  quartic  having  a  node  at  the 


10  CURVES  IN  A  PLANE. 

given  point.  One  of  the  tangents  at  the  given  point  is  perpendicular 
to  the  direction  of  the  force  acting  at  that  point,  and  the  other  makes 
with  thai  direction  an  angle  which  is  bisected  by  the  given  direction. 

Section  7.     The  Inverse  Curve. 

The  inverse  of  this  quartic  with  respect  to  the  origin  is  ob- 
tained by  the  transformation, 

«  =  ^2  I  ■  2»    /3  = 


Substituting  these  in  (26),  we  obtain  the  conic 
3(1  +  y"){My"+2<pY  -  ^rp]e  -  (^^H-  ^)(1  -  y")^v 

+  My"  -  2^l^y'  -  <prl^W  +  [12Pi(v?  +  t/V) 

-  2(3P2  +  Pi2/')(^  -  y'<pm  -  mP2y'  -  2Pi)(^  -  yV)    ^^^^ 

-  12P^y'i<p  +  2/V)]7/  +  4P1P2  =  0. 
The  discriminant  of  (28)  is 

9(1  +  yy[(<p'  -  rl^)a  -  y")  +  ^<p^y'f  >  0 

and  the  conic  is  an  hyperbola.     The  asymptotes  of  this  hyper- 
bola are  parallel  to  the  lines 

WrPy''  +  2<p'y'  -  <pyp]e  -  {<p'  +  ^2)(i  _  y'')^^ 

+  [<prh"  -  2Vy'  -  <pW  =  0. 

Hence  the  asymptotes  of  the  hyperbola  are  parallel  to  the 
tangents  to  the  quartic  at  the  origin.  It  follows  that  the  axes 
of  the  hyperbola  make  with  the  given  direction  angles  of  ±  45 
degrees. 

The  inverse  of  the  quartic  of  thereom  VI  is  an  hyperbola.  The 
asymptotes  of  the  hyperbola  are  parallel  to  the  tangents  to  the  quartic 
ai  the  origin. 

Section  8.    Determination  of  the  Conies,  Obtained  in  Accordance 

vrith  Theorem  I,  which  are  Hyperosculated  by  Their  Circles  of 

Curvature  at  the  Origin. 

The  differential  equation  of  all  the  circles  in  a  plane  is 

y'"  =  r^7-  (29) 


GEOMETRIC  PROPERTIES   OF  THE  SYSTEM.  11 

If  the  conic  is  hyperosculated  by  its  circle  of  curvature  at  the 
origin,  the  first  three  derivatives  are  the  same  for  both  curves 
at  that  point. 

For  the  conic  (10),  we  find 

fh=y'-  i=2,".  g=6(to"+w.     (30) 

Substituting  these  values  in  (29),  we  obtain 

-[Qiy"+Q2](l-hy")  =  2yy'.  (31) 

This  is  linear  in  y".  Hence  in  the  system  of  oo^  conies  cor- 
responding to  a  lineal  element,  by  theorem  I,  there  is  one  which 
is  hyperosculated  by  its  circle  of  curvature  at  the  origin. 

The  coordinates  of  the  center  of  this  hyperosculating  circle 
are 

The  locus  of  the  center  of  this  hyperosculating  circle  as  y' 
varies,  obtained  by  eliminating  y"  and  y'  between  (30)  and  (31), 
is 
{xf^y<p  -  2<pyyP)X^  +  {<py<p  +  2<p^^P  -  4^^<p  -  rP,rP)X^Y 

-  {<pyyp  -  2xl^y<p  -  xl^^rP  +  <Px<p)XY'  +  {<p,yp  -  2^x^)P     (33) 

This  is  a  cubic  having  a  conjugate  point  at  the  given  point: 
the  tangents  at  the  given  point  are  the  two  minimal  lines. 

Theorem  VII.  Of  the  oo  ^  conies  eorresponding,  in  aceordance 
with  theorem  I,  to  a  given  lineal  element,  there  is  one  which  is  hyper- 
osculated by  its  osculating  circle  at  the  given  point.  The  locus  of 
the  center  of  this  hyperosculating  circle,  as  y'  varies,  is  a  cubic  with 
a  conjugate  point  at  the  given  point.  The  tangents  at  this  point 
are  the  minimal  lines. 

Section  9.     Coordinates  of  the  Mid-point  of  the  Chord  of  the 
Hyperbola  {28)  Cut  from  the  Normal  to  the  Corresponding 
Lineal  Element. 
0   The  equation  of  the  normal  is 

^=-^^  (34) 


12  CURVES  IN  A  PLANE. 

Regarding  (28)  and  (34)  as  simultaneous  equations,  we  obtain 

3(1  +  yy{<p  +  2/V)  (^  -  y'<p)e 

The  discriminant  of  this  quadratic, 

4(1  +  y'W\^  -  2/V)Pi[Pi(^  -  2/V)  +  12P2(^  +  yV)], 
is  a  polynomial  of  even  degree  in  y',  with  the  real  linear  factor 
^  ~  y'<P'  Hence  corresponding  to  each  point  of  the  plane 
there  is  a  range  of  values  of  y'  for  which  the  discriminant  is 
>  0.  Throughout  this  range  of  values  of  y',  (34)  intersects  its 
corresponding  hyperbola  (28)  in  two  real  and  distinct  points. 
From  (35)  and  (34)  we  find  the  coordinates  of  the  mid-point 
of  the  chord  to  be 

_  2Pyy'  ^  2Px 

3(^  +  t/V)(i  +  y'')'  ^{<P^y'^){i  +  y")'    ^   ^ 

With  each  lineal  element  through  a  given  point,  we  may  as- 
sociate the  mid-point  of  the  chord  of  the  hyperbola  corresponding 
to  the  element  cut  from  the  normal  to  the  element.  The  locus 
of  this  mid-point,  as  the  lineal  element  revolves  about  the  given 
point,  obtained  by  eliminating  y'  between  equations  (36),  is 
3(<p7-^X)(Z2+  p)_2[^,y2 _  (^^ _  ^,)XY-  VyX^]  =  0.    (37) 

This  is  a  circular  cubic  with  a  double  point  at  the  given  point. 
The  real  asymptote  to  this  cubic  is  parallel  to  the  direction  of 
the  force  acting  at  the  given  point. 

Theorem  VIII.  The  inverse  of  the  quartic  of  theorem  VII  is 
an  hyperbola.  The  locus  of  the  mid-point  of  the  chord  of  this 
hyperbola  cut  from  the  normal  to  the  lineal  element,  as  the  direction 
of  the  element  varies,  is  a  circular  cvhic  with  a  double  point  at  the 
given  point.  The  real  asymptote  of  the  cubic  is  parallel  to  the 
direction  of  the  force  acting  at  the  given  point. 

Section  10.     Curves  of  the  System  which  Have  Five-Point  Contact 
with  Their  Osculating  Circles. 

The  differential  equation  of  all  circles  in  the  plane  is 

!'"'  =  f$7-  (38) 


GEOMETRIC  PROPERTIES  OP  THE  SYSTEM.  13 

DiJBFerentiating  this,  we  have 

y    '- (1+7^2+ r+p.  (38) 

If  a  curve  has  five-point  contact  with  its  osculating  circle, 
the  first  four  derivatives  at  the  point  of  contact  are  the  same  for 
both  curves.  Substituting  the  values  of  y'"  and  y^  from  (38) 
and  (380  in  the  differential  equation  of  the  system  (8),  we  have 

[r|^2+^4]2/"+Q5=0.  (39) 

Hence  corresponding  to  each  lineal  element  there  is  one  curve 
of  the  system  which  has  five-point  contact  with  its  circle  of 
curvature.    If  we  eliminate  y'  and  y"  between 

X^-^r^,    Y^'-±^  (39') 

y  y 

and  (39),  we  get  the  locus  of  the  centers  of  these  five-point  circles, 
as  the  direction  of  the  element  varies,  to  be 

^xP  -  {<Py  +  ^.)XY  +  4'yX'  +m+  <pX)  =  0.     (40) 

This  is  a  conic  passing  through  the  given  point.  Its  tangent 
at  this  point  is  perpendicular  to  the  direction  of  the  force  acting 
at  the  point. 

Theorem  IX.  Of  the  oo^  curves  of  the  system  corresponding  to 
each  lineal  element,  there  is  one  which  has  five-point  contact  with 
its  osculating  circle  at  the  given  point.  The  locu^  of  the  centers  of 
these  circles,  as  the  direction  of  the  element  varies,  is  a  conic  passing 
through  the  given  point.  Its  tangent  at  this  point  is  perpendicular 
to  the  direction  of  the  force  acting  at  the  point. 


CHAPTER  III. 

Conversion  of  the  Properties  Stated  in  Theorems  I  to  IX. 

The  quadruply  infinite  system  of  curves  defined  by  (8)  pos- 
sesses the  properties  stated  in  the  theorems  I  to  IX.  We  now 
consider  the  converse  question,  if  a  quadruply  infinite  system  of 
curves  possesses  these  properties,  does  there  exist  a  field  of  force 
such  that  the  pressure  on  any  curve  of  the  family  is  some  constant 
multiple  of  the  normal  component  of  the  acting  force? 

Section  1.     Conversion  of  Property  I. 

The  differential  equation  of  any  oo*  family  of  curves  in  the 
plane  is  y^  =  G{x,  y,  y',  y",  y'"),  when  G  is  an  arbitrary  function 
of  its  arguments. 

If  this  family  possesses  property  I,  the  centers  of  the  osculating 
conies  of  the  oo  ^  curves  of  the  family  defined  by  any  curvature 
element  {x,  y,  y',  y")  lie  upon  a  conic  through  the  point  {x,  y) 
in  the  direction  y'. 

The  most  general  equation  of  such  a  conic,  referred  to  the 
point  {x,  y)  as  origin,  is 

Ayo?  +  A^a^  +  Az^  +  {ay'  -  ^)  =  0,  (41) 

where  A\,  A2,  Az  are  arbitrary  functions  of  x,  y,  y',  y". 
The  coordinates  of  the  center  of  the  osculating  conic  are 

^y"y"'  o    ^y'y"y"'  -  %'" 


^  K'l.f'f^  Q-..".iiIV '        P 


by'"^  -  Sy'Y"^'     '^  ~  5y"'^  -  Sy"y^  ' 

Substituting  these  in  (41),  we  have  after  reduction 

y"^  =  B^y"" -\- B^y'"  +  Bz,  (42) 

where  the  B's  are  arbitrary  functions  of  x,  y,  y',  y" .  Hence  if  a 
system  of  oo^  curves  of  the  plane  has  property  I,  its  defining 
equation  must  be  of  type  (42). 

Section  2.     Conversion  of  Property  II. 
We  now  find  those  systems  of  type  (42)  which  have  property 

II. 

14 


CONVERSION  OF  PROPERTIES.  15 

The  locus  of  the  centers  of  the  o?^  osculating  conies  of  the 
system  (42)  corresponding  to  a  curvature  element  {x,  y,  y',  y") 
is  obtained  by  substituting 

Making  this  substitution,  we  obtain 

[Iby'"  -  ^B,y"'  -  3B2y'y  -  B^y^W 

(43) 
+  m^y"  +  2Bzy')hk  -  Bzk^  -  Sy'yh  +  dy"k  =  0. 

The  curvature  K  of  this  conic  is 

2[3B^y''  -  5]y" 

If  the  system  (42)  has  property  II,  then 
2[SBiy"  -  5]y"  2y" 


from  which,  Bi  =  2ly". 

The  differential  equation  of  all  quadruply  infinite  systems 
having  properties  I  and  II  is  then 

y"^  =  ^rV'"  ■\- B^y"  +  Bz,  (44) 

when  Bz  and  ^3  are  arbitrary  functions  of  x,  y,  y',  y". 

Section  3.     Conversion  of  Property  III. 

The  conic  corresponding,  in  accordance  with  property  I,  to  a 
curvature  element  of  the  system  (44),  may  be  obtained  from  (43) 
by  replacing  Bi  by  2jy".    This  gives 

W  +  3B,y'y  +  Bzy")h?  -  {ZB^y"'  +  2Bzy')}ik 

+  Bzk  +  Zy'yh  -  Zy"'k  =  0. 
For  this  conic,  we  have 

d^k  ^„    „      d^k      „  „      dh 

^z=-^B,y,     ^,=  2y,     ^=y. 

Substituting  these  in  the  formulas  (11)  for  the  second  center 
of  curvature, 


16  CURVES  IN  A  PLANE. 


(45) 


4(1  +  y'')/      6(1  +  yyB2 

^  (1  +  y'^jl  -  Sy")     6(1  +  y")y'B2 
2/2  2y"  8y"^ 

If  system  (44)  has  property  III,  the  coordinates  of  the  second 
center  of  curvature  obtained  in  (45)  must  satisfy  the  equation 
of  a  parabola  passing  through  the  given  point  with  its  axis 
parallel  to  the  given  direction. 

The  most  general  equation  of  such  a  parabola  is 

(2/2  -  y'x^y  +  Ciy2  4-  C2X2  =  0,  (46) 

when  X2,  yz  are  the  coordinates  of  any  point  on  the  parabola 
referred  to  the  given  point  (x,  y)  as  origin,  and  C\,  C2  are  arbi- 
trary functions  of  x,  y,  and  y'. 
Substituting  the  values  from  (45)  in  (46),  we  obtain 

Solving  this  for  B2,  making  a  slight  change  in  notation,  we 
obtain 

B2  =  Hiy"-{-H2, 

where  Hi  and  H2  are  arbitrary  functions  of  x,  y,  3^. 

All  the  quadruply  infinite  systems  having  properties  I,  II, 
III  are  then  defined  by  the  differential  equation 

2/"^  =  ^  y'"^  +  ili^y"  +  H2)y'"  +  Bz,  (47) 

where  Hi  and  H2  are  arbitrary  functions  of  x,  y,  y'  and  B%  is  an 
arbitrary  function  of  x,  y,  y' ,  y" . 

Section  4-  Conversion  of  Property  IV. 
In  order  to  convert  Theorem  IV,  we  state  it  in  a  form  which 
does  not  assume  the  existence  of  a  force,  as  follows:  At  each  point 
{x,  y)  of  the  plane,  there  exists  a  certain  direction  of  slope  co(x,  y) 
svxih  that  the  sine  of  twice  the  angle  this  direction  makes  with  the 
given  element  is  equal  to  three  times  the  tangent  of  the  angle  the  given 
element  makes  with  the  parabola  corresponding  to  it  by  theorem  III. 


CONVERSION  OF  PROPERTIES.  17 

The  parabola  corresponding,  in  accordance  with  property  III, 
to  the  00  2  curves  of  system  (47)  passing  through  a  given  point 
{x,  y)  in  a  given  direction  y',  is  found  by  eliminating  y"  between 

4(l  +  y^V      6(1 +  y-^)'^ 
^2  = 2y^' '^^ —  ^^'^y    +  ^2;» 

(1  +  y") ,,     .  „,     6(i  +  y-W(giy"  +  g2) 

2/2  =      22/"      ^^  ~  ^^  ^ W"  * 

Carrying  out  this  elimination,  we  obtain 

6^2(2/2  -  yW)  +  (1  +  2/'=')[3(l  +  y")Hr  +  82/']z/2 

+  (1  +  2/")[2  -  W'  -  3Fi(l  +  y")y']x2  =  0. 

The  slope  of  the  tangent  to  this  parabola  at  the  given  point  is 

_2-Qy"-SHra-^y")y' 
3(1  +  y")Hi  +  8y'       ' 

If  the  system  (47)  has  property  IV,  we  have 


2(0,  -  y') 


of   'J  2  -  6y^^  -  3gi(l  +  y- yi 

'^L^-^       S{l-\-y")Hi-\-Sy'       J_ 

[2  -  6y-'  -  Bffid  +  y")y']y'  fco-y- T' 

^  3(1 +  2/'=^)^!+ 8/  ■^Ll  +  coy'J 

Solving  this  for  iTi,  we  find 

1  +  ic"^  2y' 


Hi 


(CO  -  2/')(l  +  W)      1  +  2/"' 
where  w  is  an  arbitrary  function  of  x  and  2/- 
The  differential  equation 

defines  all  the  quadruply  infinite  systems  having  properties  I, 
II,  III,  IV. 

Section  5.     Conversion  of  Property  V. 

Property  V  may  be  stated  in  a  form  that  does  not  require  the 
existence  of  a  field  of  force  by  replacing  the  direction  of  the 
acting  force  by  the  function  o){x,  y)  defined  in  Section  4. 

The  most  general  conic  fulfilling  the  conditions  of  theorem  V  is 

Gia^  +  GiaP  +  Gs8''  +  fi  -  ma  =  0,  (49) 


18  CURVES  IN  A  PLANE. 

where  the  G's  are  arbitrary  functions  of  x,  y,  and  y',  and  m  is 
determined  from  the  relation 


m 


m 


^Ll  +  co/J 

(1  +  co2)(l  +  y")y'  +  2(co  -  y'){\  +  cot/') 


to  be 


(1  +  y")0.  +  d")  -  2(co  -  y'){l  +  coy')2/'* 

The  center  of  the  conic  corresponding,  according  to  property 
I,  to  the  system  (48)  is 

Z{H,y"+my" 
a  =  — 


where 


S{Hiy"  +  H,)Y^ - 

453' 

3(22/"  +  H^y'y  +  H,y')y" 

SiH^y"  +  ^2)Y' 

-4^3 

1  +  0)2 

22/' 

(co-2/')(l  +  <^y')      1  +  2/''* 
By  property  V,  these  satisfy  (49).     Substituting,  we  have 
Liy'"  +  L2y"' +  L,y" 

+  2F2(co  -  2/0(1  +  co2/')[(3Fiy"  +  H^Yy"  -  4^3]  =  0, 

when  the  L's  are  functions  of  Gu  G2,  G3,  H2,  x,  y,  and  y'  and  there- 
fore arbitrary  functions  of  x,  y,  and  y'. 
Solving  this  for  Bz, 

Bz=  Miy'"-\-M,y'"+Mzy", 

when  the  M's  are  arbitrary  functions  of  x,  y,  and  y'. 

Hence  all  systems  possessing  properties  I,  II,  III,  IV  and  V 
have  the  form 

2^     -y"^'      +iL(l  +  co2/0(co-2/')      l+2/'^J^+^^P       (50) 

+  Miy"'+M2y"'+Mzy". 

Section  6.     Conversion  of  Property  VI. 
We  state  theorem  VI  without  assuming  the  existence  of  a 
force  by  replacing  the  direction  of  the  force  acting  at  the  given 
point  by  the  function  (a{x,  y)  defined  in  Section  4. 


CONVERSION  OF  PROPERTIES.  19 

We  determine  the  slopes  mi  and  mi  of  the  tangents  to  the 
quartic  of  theorem  VI  from  the  conditions 

1        (a—  y'       y'  —  tni 


TOl  =   — 


CO 


1  +  C02/'      1  +  2/'wi2  * 


(51) 


(52) 


Solving  the  second  of  these  equations  for  m2,  we  find 

2y^+coy^'-co 

We  now  find  the  locus  of  the  foci  of  the  hyperosculating 
parabolas  of  the  curves  of  (50)  which  correspond  to  a  given  lineal 
element  (ar,  y,  y').  It  is  obtained  by  substituting  the  values  of 
y",  y"',  y"  from  (22)  and  (25)  in  (50).    The  result  is 

m^{c?  +  ^f  +  {2[M2(1  +  2/")  +  6ilf  ii/']/3 

+  2[3F2(1  -  y'^)  -  Miy\\  +  y'^)\a\ {o?  +  ^) 

+  [121/'='  +  m^y'iX  +  y")  +  M  i(l  +  y'^X^  (53) 

+  [12t/'(l-2/'^)-2Mi2/'(l+2/'^)2+3Fi(l+2/'=')(l-3y"')]a/3 

+  [3(1  -  y'^)  +  iWi2/''(l  +  y'''?  -  ^H^y'a  -  y")W  =  0, 

where 

l  +  co^^ 2y[_ 

^'      (CO  -  2/0(1 -I- coi/O      l  +  2/'='* 

This  is  a  bicircular  quartic.  Hence  if  an  qo*  system  has 
properties  I  to  V,  it  also  has  the  property  that  the  foci  of  the 
hyperosculating  parabolas  of  the  curves  corresponding  to  a 
lineal  element  lie  upon  a  bicircular  quartic. 

In  accordance  with  theorem  VI,  the  slopes  of  the  tangents  to 
(53)  at  the  given  point  must  equal  mi  and  m2  of  (51)  and  (52). 

Forming  the  product  of  these  slopes,  we  have 

3(1  -  y")  +  M^y'W  +  y^  -  SH^y'd  -  y") 


12y"  +  QHiy'a  +  y'')  +  Mi(l  +  y'Y 

_  _  2y'  +  <^y'^  1  -  CO 
co(l  +  2co2/'-i/'^)- 
This  solved  for  Mi  gives 

M,=       ^  32/'(l  +  co^) 


(54) 


1  +  2/''      (co-2/')(l  +  co2/0(l  +  2/'')- 
The  differential  equation  of  all  the  systems  having  properties 


HI 

(55) 


20  CURVES  IN  A  PLANE. 

I,  II,  III,  IV,  V,  VI  is  of  the  form 

when  Hi,  If 2,  Mi  are  arbitrary  functions  of  x,  y,  and  y',  and  w 
is  an  arbitrary  function  of  x  and  y. 

Section  7.     Conversion  of  Property  VII. 

The  most  general  cubic  having  the  properties  required  by 
theorem  VII  is 

TiX^  +  TiX^Y  +  TzXY'  +  T,r+X^+Y^  =  0,     (56) 

when  the  7"s  are  arbitrary  functions  of  x  and  y. 

The  conies  of  theorem  I  corresponding  to  system  (55)  may 
be  obtained  from  (10)  by  replacing  Qi,  Q2,  Q3,  Qi,  Qh  by  Hi,  H2, 
Ml,  Mi,  Mz  respectively.  Hence  dkjdh,  (Phjdh^,  d^kjdh?  of  the 
conic  may  be  obtained  from  (30)  by  replacing  Qi  and  Q2  by  Hi 
and  Hi  respectively.    We  find 

dh        ,      d?k      ^  ,,      d?k  ^,tx    »   ■    ttv   .. 

Substituting  these  in  the  differential  equation  of  all  the  circles, 

y     1  +  2,'*' 

we  have,  after  substituting  for  ^1  its  value  in  (53), 

r         l  +  co^  ^y'    1  „  ,   „  _       2yy^ 

lio>-y'){\  +  <.y')-l  +  y''\y    +  ^^  "  "  1  + y'-     ^^7) 

In  accordance  with  property  VII,  if  we  eliminate  y'  and  y" 
between  (57)  and  equations  (32),  we  get  a  cubic  of  the  form 
(56). 

Substituting  in  (57)  the  values  y'=  -  (Z/F),  2/"  =  (Z'+  P)/l^, 
and  denoting  by  Hi  the  function  of  X  and  Y  which  Hi  becomes 
on  replacing  y'  by  —  (X/Y),  we  obtain 

g,'[^^"^|j^>J-"^J  +  X^+P=0.        (58) 


CONVEESION  OP  PROPEKTIES.  21 

In  order  that  (58)  may  have  the  form  (56),  we  must  have 
^2'  r  Y(^y+^)(Y-o>X)-\  ^  y^^3_^  T2X'Y+  TzXY'+  T,Y\ 

_  iTiX^+  T2X^Y+  TzXY'+  rP)(l  +  oy") 
>^^  "  F(a)y4-X)(F-coZ) 

(a,+  f)(l-cof) 


^2'  = 
Hence, 


Tj  _  K,y"+K,Y"+KzY'  +  K,  .^^. 

^'  -  (0,-2/0(1  +  0,/)  '  ^^^^ 


when  the  K's  are  arbitrary  functions  of  x  and  y. 

The  differential  equation  defining  all  00  ^  systems  having  the 
first  seven  properties  is 

2/^  =  4  y""  +  [Hiy"  +  my'"  +  M^y"'  +  M,y"'  +  ilf 32/",    (60) 


where 


_  1  +  0,2  22/' 

^'  ~  (0,  -  2/0(1  +  0,2/')  ~  fTF' 

__       K,y"-^K,y''+K^y'  +  K,  .^^^ 

H2  -         («_  2,0(1 +0,2/0         '  ^^^^ 

^  3  32/'(l  +  o,2) 


1  +  2/"      (o,-2/)(l  + 0,2/0(1 +  2/'*)' 

Xi,  X2,  K3,  Ki  are  arbitrary  functions  of  x  and  2/  and  M2,  Mz 
are  arbitrary  functions  of  x,  y,  and  2/', 

Section  8.     Conversion  of  Property  VIII . 

We  find  the  quartic  corresponding,  in  accordance  with 
theorem  VI,  to  (60)  by  substituting  in  (53)  the  particular  values 
of  Hi,  H2,  and  Mi  given  in  (61).  The  inverse  of  this  quartic  is 
the  hyperbola 


22  CURVES  IN  A  PLANE. 

[3(1  -  y")  +  Miy'W  +  y^  -  3H^y\l  -  y'*)]^ 
+  [Wil  -  y")  -  2Miy'(l  +  yy 

+  SH,{1  +  y'Yl  -  Sy")]^V 

+  [W  +  6H^y'(l  +  y")  +  Mi(l  +  y'')]^^  (62) 

+  2[M2il  +  y")  +  m,y']r} 

+  2[3Zr2(l  -  y")  -  M,y'{l  +  2/'^]^  +  ^M,  =  0, 

where  Hi,  H2,  and  il/i  have  the  values  given  in  (61). 

We  find  the  coordinates  of  the  mid-point  of  the  chord  of  this 
hyperbola  cut  from  the  line  v  =  —  (1/2/0^  to  be 

_  2M,y'      2H,y" 
^~      3     '^l  +  y''' 

(63) 
2M2       2H2y' 


Y=  - 


1  +  y'" 


In  accordance  with  theorem  VIII  (which  we  restate  by  re- 
placing the  direction  of  the  force  acting  at  the  given  point  by 
CO  (a:,  y))  these  coordinates  must  satisfy  a  cubic  of  the  form 

(Z2  +  Y^)io}X  -  Y)  +  AX^+  BXY  +  CP  =  0.      (64) 

Substituting  the  values  of  X  and  Y  from  (63)  in  (64),  we  obtain 

[^  +  ff^]  (1  +  /')(1  +  «/)  +  ^^"  -  By'+  C  =  0. 

Solving  this  for  if 2  and  introducing  Ni  =  —  ^A,  N^  =  }5, 
Nz=  —  f  C,  we  obtain 

^  ^Niy^+N^y'+Nz        SH^y' 


(1  + 2/'=')(l  +  coy')      a  +  y")' 

where  «  and  the  N*s  are  arbitrary  functions  of  x  and  y,  and  Hi 
has  the  value  given  in  (61). 

The  differential  equation  of  all  00  *  systems  possessing  the  first 
eight  properties  is 


f'-Y'^"" 


CONVERSION  OF  PROPERTIES.  23 

"^lL("-2/')(H-coy')      l-\-y"V' 

^     (CO -2/0(1+ cot/')     \y 


+ 


r 

L  (1  +  2/'')(l  +  CO2/0 

(co- 2/0(1  + W2/0  J 

when  the  iiC's,  the  N's,  and  co  are  arbitrary  functions  of  x  and  y, 
and  ilf  3  is  an  arbitrary  function  of  x,  y,  and  2/'. 

Section  9.     Conversion  of  Property  IX. 
We  obtain  the  curve  of   system   (65),   corresponding  to  a 
given  lineal  element,  which  has  five  point  contact  with  its  os- 
culating circle,  from  (39)  by  replacing  Q2,  Q4,  and  Q5  by 

Kxy"  +  K<,y"  ^  K^y' +  K, 


H2  = 
if2  = 


(<0  -  2/0(1  +  C02/0 


(l  +  2/'')(l  +  co2/0      1  +  y" 
and  ikfa  respectively.    The  result  is 

(1  +  2,")(1  +  .,/)  ^    +il/3-0. 
Hence 

_      il/3(l  +  y'^W  +  1)  .„, 

y    -        N,y"+N,y'  +  N,  '  ^^' 

The  center  of  the  corresponding  osculating  circle,  obtained 
by  substituting  this  value  of  y"  in  (390,  is 

y_y'[Niy"+N2y'  +  Nz] 

Mzil  +  C02/O 

(67) 

_N^y;^_±N2y^±Nz 

Mzil  +  a;2/0        • 


24  CUKVES  IN  A  PLANE. 

In  accordance  with  property  IX  (restated  without  assuming 
the  existence  of  a  force),  these  coordinates  satisfy  a  conic  of 
the  form 

EiZ2  -  IkXY  +  Rzr  +  (X  +  wF)  =  0,  (68) 

when  Ri,  Rz,  R3  and  w  are  arbitrary  functions  of  x  and  y. 
Substituting,  we  obtain 

[Ni  Y"  +  N^y'  +  N,][R^y"  +  R,y'  +  Rz] 

+  iy'  -«)(!  +  o}y')M,  =  0. 
Solving  for  Mz, 

^  _  [Njy''  4-  Njy'  +  NzWR^y"  +  Riy'  +  Rz] 
(CO -2/0(1  + "2/') 
when  the  iJ's  are  arbitrary  functions  of  x  and  y. 
Hence  the  systems  of  type 

IV  _  A   n,7A__l±jl 2y^l    u 

y    y"y  "^L(«- 2/0(1 +"2/0   i  +  y"r 

TKiy"^-K,y"■\-Kzy'  +  K,■^   ,„ 
■^L         (CO  - /)(1  +  C02,')         J 2^ 

4.  r_J 3y-(l  +  co^) 1 

^  L 1  +  2/''      (1  +  2/'0(co  -  2/0(1  +  C02/O  J  ^ 

0 


■^L(l  +  2/")(l  +  co2/0 


(69) 


3(ii:iy^*  +  ^22/^'  +  Ji^ay^'  +  T^^v'  1   , 
+  2/'0(«- 2/0(1 +  C02/O      J^' 


+ 


(1 
l^^y'^  +  i\^22/'  +  iV^3][/?i2/''  +  ^^22/'  +  M 


\y". 


(CO- 2/0(1 +  C02/O 

where  co,  iVi,  iVz,  ^3,  ^1,  -K^2,  -K^s,  -K^4,  i^i,  jR2,  and  Rz  are  arbitrary 
functions  of  x  and  2/,  are  completely  characterized  by  properties 
I,  II,  III,  IV,  V,  VI,  VII,  VIII,  IX. 


CHAPTER  IV. 

Complete  Chaeacterization. 

Types  (69)  and  (6)  involve  the  derivatives  y',  y",  y'",  y^ 
in  exactly  the  same  way;  but  (69)  is  more  general  than  (6)  with 
respect  to  the  arbitrary  functions  of  x  and  y  appearing  in  the 
coefficients. 
Section  1.    Relations  Existing  among  the  Arbitrary  Functions 

Entering  into  the  Differential  Equation  of  the  System  of  Curves 

Having  Properties  I-IX  if  it  is  to  be  of  the  Special  Form  given 

in  Chapter  I. 

We  now  determine  the  relations  that  must  exist  among  these 
arbitrary  functions  in  order  that  (69)  may  be  of  the  special 
type  (6). 

By  direct  comparison  of  (69)  and  (6),  we  have 


<a 

Y 

(70) 

Ni 

_<Py        ^^^<P'-^v        ^               >Px 
<P                             <P                                    <P 

(71) 

Ri 

^i              T,               ^X-h    (Pv              Jf                <PX 

<P                                  <P                                   <P 

(72) 

2(Py\l/    —     \l/y<P                ^^                  (py(p    "     xf/ x<P    "     ^y^    +     '^'(pX^ 

ipyip  —  2\py<p  +  (Pj,(P  —  yffgxf/       ^^        (p^xp  —  2\l/x(p 

^2                                  >       ^^4-                 ^2 

(73) 

itroc 

lucing  (f>  =  log  <f)  and  substituting  xp  —  aj0,  we  have 

Ni 

=     <t)y,          Ni    =     <f>x    —    W0y    —    Uy,          Nz    =     —    03x    —    Oi4>x, 

(74) 

Ri- 

=    Ul^y  +    COj,,          JB2    =    01/  +    (^4>X  +    i>ix,          RZ    =    <f>X, 

(75) 

Ki  =   W0I,  —  COy,       iL2  =   (1  —  W  )0J/  +  W0a;  —  COs  —  C0CO«, 

-         -  -       (76) 

Kz=  {\  —  (»)^)<f>x  —  oi(t>v ~  2a)y  —  cowx,     i^4=  —2(ax—oi(f>x' 

From  the  first  and  third  equations  of  (74),  we  have 

25 


26  CUEVES  IN  A  PLANE. 

<i>y=Nr,    <!>:,=  --^-5.  (77) 

CO 

Substituting  these  values  in  the  second   equation  of  (74),  it 
becomes 

wWi  +  U)N2  +  Nz+0}^+  UUy  =   0.  (78) 

From  (77)  we  have 

(78)  and  (79)  are  necessary  conditions.  To  show  that  they  are 
also  suflBcient:  if  (79)  holds,  we  can  find  a  function  (f>  which 
satisfies  both  of  the  equations  (77).  Then  on  account  of  (78), 
equations  (74)  hold.  Finally  (70)  and  <f>  =  log  <f>  determine  a 
pair  of  functions  <f>  and  ^  in  terms  of  which  we  can  express  Ni, 
Nz,  Nz  and  u  in  the  forms  given  in  (70)  and  (71). 

From  (75)  and  (77),  we  have 

Ri  =  oiNi  +  oiy,    R2=Ni-N3,    Rs=  -^^+^",      (gQ) 

CO 

and  from  (76)  and  (77), 

Kl  =   CoiVi  —  C0„,       ^2  =    (1  —  CO^)Ni  —  N3—  2cOx  —  COCOy, 
__  (1   —   CO^)   .^  ...  0)x  „  rr  HT  ^^^^ 

K3=  - -Nz  -  coiVi 2uy,    Ki=  Nz-  cox. 

CO  CO 

If  relations  (80)  and  (81)  hold,  we  can  express  Ri,  R2,  Rz, 
Kl,  K2,  Kz,  and  K^  in  terms  of  (p  and  \J/  in  the  forms  given  in 
(72)  and  (73). 

We  now  derive  relations  equivalent  to  (80)  and  (81)  which  admit 
of  geometric  interpretations. 

From  (75)  we  have  0x  =  Rz,  <f)y  =  {Ri  —  cS)yfo). 

Substituting  these  in  the  second  equation  of  (75),  it  becomes 

ui^Rz  —  <aR2  -{-  Ri  —  o}y-{-  coco*  =  0.  (83) 

From  (80), 

Ri  =  coiVTi  +  CO,,    Rz=  -  ^^^^^t-^.  (83') 

CO 

The  equations  (83)  and  (83')  are  equivalent  to  the  equations  (80). 
From  (76)  we  have 

_       Ki  +  o}y      -  Ki  +  2co» 


COMPLETE  CHABACTERIZATION.  27 

Substituting  these  values  in  the  second  and  third  equations 
of  the  set  (81),  we  obtain 

(1  -  o)^)Ki  -  o)Ki  -  UK4  +  uy-  2co2aj„  -  Scoco^  =  0,     (84) 
uKi  +  ojKs  +  (1  -  o}^)K4,  +  2co,  -  oi'u}^  +  3ajco„  =  0.     (85) 

Multiplying  (85)  by  w  and  adding  to  (84),  we  obtain 
Ki  -  coiiTz  +  oi^Kz  -  o)'K,  +  (coj,  -  (acc^Kl  +  w2)  =  0.    (86) 

Multiplying  (84)  by  co  and  subtracting  from  (85),  we  obtain 

u^Ki  +  03^K2  +  uKz  +  Ki  +  2{o3^  +  «a)„)(l  +  <a^)  =  0.     (87) 

From  (82)  we  have 

Ki  =  uNi  -  o3y,    Ki=  N3-  w,.  (88) 

Relations  (86),  (87),  and  (88)  are  equivalent  to  relations  (82). 

Section  2.     Geometric  Interpretations  of  the  Relations  of  Section  1. 

We  now  obtain  geometric  interpretation  of  relations  (78), 
(79),  (83),  (86),  (87),  (88). 

Interpretation  of  {78). — The  radius  of  curvature  of  the  bi- 
circular  quartic 

(^2  +  ^2)2  _^  ^a<x  +  6/3)  (q:2  +  ^2)  +  ca2  +  ^^^  +  e/32  =  0 

at  the  origin  is 

{l-\-^"yi\d+2e^') 

^=       W+W)      •  ^^^^ 

If  in  the  bicircular  quartic  (26)  we  take  y'  —  rp/cp,  the  slopes 
of  the  tangents  at  the  origin  are  j8'  =  \l//(p  and  /3'  =  —  (p/\l/. 

We  now  apply  formula  (91)  to  the  branch  of  this  quartic  for 
which  j3'  =  \l/(<p.  We  obtain,  after  replacing  the  direction  of 
the  force  acting  at  the  given  point  by  the  (a(x,  y)  considered  in 
Section  4,  Chapter  III, 

1               (1  +  aj2)3/2 
R^lr^. 1  T— n-  (92) 


£0 CO''  I 


Substituting  in  (92)  ^xlf  =  —  Nz,  {ipx  —  ^v)l^  =  -^2, 
<pj<p  —  Ni  from  (71),  we  obtain  the  radius  of  curvature  of  the 
quartic  corresponding  to  (69),  by  theorem  VI,  for  2/'  =  \A/v  and 


28  CURVES  IN  A  PLANE. 

/3'  =  ^jip.    This  gives 

1        (1  +  oiy^ 

Consider  now  the  lines  of  force  defined  by, 

y'  =  u{x,  y). 

The  radius  of  curvature  of  the  curve  of  this  family  passing 
through  the  given  point  is 

From  (93),  (94),  and  (98)  we  obtain 

4E  =  pi.  (94') 

Theorem  X.  The  slopes  of  the  two  branches  of  the  bicircular 
quartic  corresponding,  by  theorem  VI,  to  the  lineal  element  (x,  y,  w) 
at  the  given  point,  are  co  and  —  (l/w).  The  radius  of  curvature  of 
the  branch  of  slope  o)  is  ^  of  the  radius  of  curvature  of  the  line  of 
force  passing  through  the  given  point. 

To  each  point  0  of  the  plane  there  corresponds,  by  theorem 
VIII,  a  definite  cubic  passing  through  the  given  point.  From 
(64)  and  (64')  we  obtain  the  cubic  of  69  to  be 

3(^2  +  P)(a,Z  -  7)  -  2[iViZ2  -  N^XY  +  N^Y^]  =  0.    (95) 

The  line  Y  =  —  1/coX  meets  this  curve  in  the  point  N, 
whose  coordinates  are 

2  (Ni  +  (cN2  +  coW3)co  2  (Ni  +  uN^  +  C0W3) 

^3  (l  +  w2)2  '     ■'^      3  (l  +  «2)2 

The  distance 

3      (1  +  «2)3/2     •  Ky^) 

From  (96),  (94),  and  (78),  we  have 

iON  =  -.  (97) 

Pi 

This  relation  is  equivalent  to  the  relation  (94'). 

Theorem  XP.  The  cubic  associated,  by  theorem  VIII,  with  any 
point  0  intersects  the  line  through  0  perpendicular  to  its  asymptote 
at  a  distance  equal  to  ^  of  the  curvature  of  the  line  of  force  passing 
through  0. 


COMPLETE  CHARACTERIZATION.  29 

Interpretation  of  (79). — The  intercepts  of  the  cubic  (95)  are 

O    CO 

From  these  we  have 

Ni=iu)'  OA,    N3=  -  iOB,  (98) 

Substituting  these  values  in  (79),  it  becomes 

^(a,-0^)-^(-j  +  3        '^        '=0.        (99) 

If  through  0  we  draw  OF  in  the  direction  u{x,  y) 
and  let  A'  and  B'  denote  the  points  at  which  this 
line  meets  the  perpendicular  to  the  axes  through 
the  points  A  and  B  respectively,  (99)  may  be 
written 

S  d  2i  cocOt»  —  OirijOti 

ai  t-^^')  -  a-y  f-^^')  +  3  -^V^  =  "•  "««) 

This  is  an  intrinsic  property,  since  it  is  true  for  any  choice  of 
rectangular  axes. 

Theorem  XI.  When  the  point  0  is  moved,  the  cubic  associated 
ivith  it  by  theorem  VIII  changes  in  the  following  manner.  Take 
any  two  fixed  perpendicular  directions  for  the  x  direction  and  the 
y  direction;  through  0  draw  lines  in  these  directions  meeting  the 
cubic  again  in  the  points  A  and  B.  Construct  the  tangent  to  the 
line  of  force  at  0.  At  A  draw  a  line  parallel  to  the  Y  axis  meeting 
this  tangent  in  the  point  A'  and  at  B  draw  a  line  parallel  to  the  x 
axis  meeting  the  tangent  in  B'.  Then  the  distance  AA'  and  BB' 
and  the  slope  a  of  the  line  of  force  satisfy  relation  (100). 

Interpretation  of  (83). — The  equation  of  the  conic  correspond- 
ing to  (69)  by  theorem  IX  may  be  obtained  from  (40)  by 
substituting 

R3. 

Making  these  substitutions,  the  result  is 

RiX'  -  R2XY  +  EsF^  +  (coF  +  Z)  =  0.  (101) 

The  line  Y  =  coX  meets  this  conic  in  the  point  M  whose  coor- 
dinates are 


—  /il. 

—   it2, 

<Px 

<p 

<P 

<P 

30  CURVES  IN  A  PLANE. 


y_  1  +  co^  „_  (l  +  «> 

■A    —    —    rt  ni         or>t        /    —    — 


ill- coi^+ w2^3*  ill  -  coilz  +  w'iJi* 

The  distance 

<'^  =  B^r^-i;«;-  (102) 

Now  consider  the  orthogonal  trajectories  of  the  lines  of  force 
defined  by 

y'=--r—\'  (103) 

^  co(a:,  y) 

The  radius  of  curvature  of  the  one  of  this  family  passing 
through  the  given  point  is 

pi  = .  (104) 

From  (102),  (104)  and  the  first  relation  of  (83),  we  have 

OM  =  p2.  (1040 

Theorem  XII.  Of  the  oo^  curves  of  the  system  (6)  which  pass 
through  a  given  point  in  a  direction  normal  to  the  direction  of  the 
force  acting  at  the  point,  there  is  one  which  has  five  point  contact 
with  its  osculating  circle.  The  radius  of  curvature  of  this  element 
is  equal  to  the  radium  of  curvature  of  the  orthogonal  trajectory  of  the 
lines  of  force  which  passes  through  the  given  point. 

Again,  consider  the  radius  of  curvature  of  the  quartic  of 
theorem  VI  corresponding  to  the  lineal  element  [x,  y,  —  (1/co)] 
for  the  branch  of  slope  —  (1/w).    By  formula  (91)  we  obtain 

(1  +  w2)3/2 
R  =  -r V, Tn  •  (105) 


—  w* w  H I 

<p  <p  <p  J 


Substitute  Rz  =  <Pxl<p,  R2  =  i<Pu  +  ^x)l<p,  Ri  =  ^vf<p  in  (105), 
to  get  the  radius  of  curvature  of  the  quartic  corresponding  to 
(69)  for  y'  =  -  1/w  and  /3'  =  -  1/w.    This  gives 

(1  +  0)2)3/2 

^  "  4(ili  -  0iR2  +  o^'Ri)  '  ^^^^^ 

From  (104),  (106)  and  the  first  relation  of  (83),  we  find 

4J?  =  P2. 
This  relation  is  equivalent  to  the  relation  (104'). 


COMPLETE  CHAEACTEEIZATION.  31 

Theorem  XIP.  The  slopes  of  the  two  branches  of  the  bicircular 
quartic  corresponding,  by  theorem  VI,  to  the  lineal  element 
[x,y,  —  (1/co)]  are  co  and  —  1/co.  The  radius  of  curvature  of  the 
branch  of  slope  —  (1/w)  is  equal  to  ^  of  the  radius  of  curvature  of 
the  orthogonal  trajectory  of  the  lines  of  force  which  passes  through 
the  given  point. 

Interpretation  of  (83')- — The  intercepts  of  the  conic  (101)  are 

ill  il3 

Then 

_         1  _        ^ 

Substituting  these  values  and  the  values  of  iVi  and  N3  from 
(98)  in  (830,  we  have 

30,2-0^ +2 r^+a)„)  =  0,     3-OB  +  2(^-a),)=0.    (107) 

These  are  intrinsic  relations,  since  they  hold  for  any  choice  of 
rectangular  axes. 

Theorem  XIII.  The  intercepts  of  the  conic  associated,  by 
theorem  IX,  with  any  point  {x,  y),  and  the  intercepts  of  the  cubic 
associated,  by  theorem  VIII,  with  the  same  point,  satisfy  the 
relation  (107). 

Interpretation  of  (87). — The  cubic  corresponding  to  (69)  by 
theorem  VII  may  be  obtained  from  (33)  by  substituting 

\l/y<P    —     2^y^ 


<P' 


-Ki-  ^  , 

The  cubic  is 
KiX'-K2X^Y+K3XY'-K,Y'-  {X'+  F2)(l  + w^)  =  0.    (108) 
This  cubic  is  intersected  by  7  =  —  (l/w)X  at  the  point  Mi 


32  CURVES  IN  A  PLANE. 

whose  coordinates  are 

^  ^  i(l  +  C02)2C0 


Y=  - 


(1  +  c^r 


The  distance 

From  (109),  (94),  and  (87),  we  obtain  the  relation 

20M  =  pi.  (110) 

Theorem  XIV.  Of  the  oo  ^  conies  of  theorem  I  which  pass  through 
a  given  point  in  the  direction  of  the  line  of  force  through  that  point, 
there  is  one  which  is  hyperosculated  by  its  circle  of  curvature;  the 
radius  of  curvature  of  this  conic  is  ^  of  the  radium  of  curvature  of 
the  line  of  force  passing  through  the  given  point. 

Interpretation  of  (86). — The  cubic  (108)  is  intersected  by 
Y  =  wX  at  the  point  M^  whose  coordinates  are 

y  = 

The  distance 

(111),  (104)  and  (86),  give  the  relation 

OM2  =  P2.  (112) 

Theorem  XV.  Of  the  00  ^  conies  of  theorem  I  which  pass 
through  a  given  point  in  a  direction  normal  to  the  direction  of  the 
line  of  force  through  that  point,  the  radium  of  curvature  of  the  one 
hyperosculated  by  its  circle  of  curvature  is  equal  to  the  radiums  of 
curvature  of  the  orthogonal  trajectory  of  the  lines  of  force  which 
passes  through  the  given  point. 

Interpretation  of  (88). — The  intercepts  of  the  cubic  (108)  are 

1+0)2  Q     I         2) 

0^,  =  ^!^^,     0B2=-'^^-^, 


^1 

-coK2-{-co'Kz-  o>^K^ 

a;(l  +  co2)2 

Kr 

-oiK2  +  o>'Kz-oi^Ki 

(1  +  0)2)5/2 

COMPLETE   CHAKACTERIZATION.  33 

Then 

rr   _  L+^'        K   --  i±^' 

Substituting  these  values  and  the  values  of  Ni  and  Nz  from  (98) 
and  (88),  we  have 


^•^^-^W'-'^^]^'- 


(113) 


This  is  an  intrinsic  relation. 

Theorem  XVI.  The  intercepts  of  the  cubic  associated,  by 
theorem  VI,  with  any  point  (x,  y),  and  the  intercepts  of  the  cubic 
associated  by  theorem  VIII  with  the  same  point,  satisfy  the  relation 
(113). 

Any  quadruply  infinite  system  of  curves  in  a  plane  along 
which  the  pressure  is  proportional  to  the  normal  component  of 
the  acting  force  possesses  properties  I,  II,  III,  IV,  V,  VI,  VII, 
VIII,  IX,  X,  XI,  XII,  XIII,  XIV,  XV,  XVI;  and,  conversely, 
if  a  quadruply  infinite  system  possesses  these  properties  (restated 
by  replacing  the  direction  of  the  acting  force  by  the  function 
o){x,  y)  defined  in  section  4,  chapter  III,  there  is  a  field  of  force  in 
which  this  system  represents  the  family  of  curves  along  which 
the  pressure  is  proportional  to  the  normal  component  of  the 
acting  force. 


VITA 

Sarah  Elizabeth  Cronin,  Bachelor  of  Science,  University  of 
Iowa,  1903;  Master  of  Science,  University  of  Iowa,  1905;  In- 
structor in  Mathematics,  Iowa  State  College,  1905-1907;  In- 
structor in  Mathematics,  Iowa  State  University,  1907-1913; 
Graduate  student  at  Columbia  University,  1913-1915. 


84 


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